$\sigma_i : [0,n] \rightarrow A$.Consider a claim of the form:

$\sigma_2$ is a logical consequence of $\sigma_1$.I think this claim is seriously underspecified, because logical consequence can only make sense given a class of interpretations of the strings.

Suppose that $A$ above is the alphabet of an interpreted language $L$. Suppose that this also determines a special class $Sent(L)$ of strings, and a class $\Sigma(L)$ of interpretations for $L$ in such that way that, for any $I \in \Sigma(L)$, and any $\phi \in Sent(L)$,

$I \models \phi$is defined, meaning,

the $L$-string $\phi$ is true in $I$.Then consider the claim,

$\sigma_2$ is a logical consequence in $L$ of $\sigma_1$.On the Bolzano-Tarski definition, this means,

$\sigma_1, \sigma_2 \in Sent(L)$ and, for any $L$-interpretation $I \in \Sigma(L)$,if $I \models \sigma_1$ then $I \models \sigma_2$.

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